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In a polynomial function like \(f(x) = x^4 - 11x^3 + 36x^2 - 16x - 64\), zeros (or roots) are the \(x\)-values that make the entire equation equal to zero. Finding zeros of a polynomial is crucial because it helps us understand the behavior and graph of the polynomial. In this exercise, we identified that 4 is a zero of the polynomial function \(f(x)\).

If you substitute \(x=4\) into \(f(x)\), you'll find that the function evaluates to zero. This means 4 satisfies the equation, confirming it as a zero. Zeros can provide valuable insights, such as where the graph of the polynomial crosses the \(x\)-axis. Identifying these points requires techniques such as factoring, the Rational Root Theorem, or synthetic division.

In more complex polynomials, like in this exercise, zeros are found through a systematic process where we use tools like synthetic division to peel away layers of the polynomial and check remainders. It's key to know that each zero can have a multiplicity, which affects how many times a graph touches or crosses the \(x\)-axis at that point.

Multiplicity in polynomials tells us how often a particular zero appears as a solution to the polynomial. For example, in the equation \(f(x) = (x-4)^3(x+1)\), the zero 4 appears with a multiplicity of 3, meaning \(x=4\) is a solution three times over.

This affects the graph, where at \(x=4\), the curve will touch the \(x\)-axis and then rebound instead of crossing through. Higher multiplicities cause poignantly different graph behaviors, like flattening at the zero. Contrary to a single root that just zips through the \(x\)-axis, a root of multiplicity two will touch and turn back, forming a tangent to the axis.

Multiplicity alters how the polynomial can be factored as well. When you factor after determining a zero and its multiplicity, you write it like \((x-4)^3\) to indicate its repeated nature. Such notations are important for conveying complete information about the behavior of the polynomial around that zero.

Synthetic division is a streamlined way to divide polynomials, especially useful for determining zeros. It is most effective when dividing by a linear expression, namely \(x - c\). This process simplifies calculations when checking zeros and their multiplicities.

In the exercise, synthetic division helped us confirm that 4 is a zero of the polynomial \(x^4 - 11x^3 + 36x^2 - 16x - 64\). We performed synthetic division three times with \(x=4\) to check its occurrence, hence verifying a multiplicity of 3.

The process of synthetic division involves using only the coefficients of the polynomial, repeatedly multiplying and adding as you shuffle them down a row. If you end with a zero remainder after the division, it affirms \(x - c\) (here \(x-4\)) as a factor of the polynomial. The final step is to analyze the resulting simplified polynomial after each division to see any further zeros, like the \((x+1)\) in our example.